Extensions to the Theory of Widely Linear Complex Kalman Filtering

TL;DR

This paper extends the theory of widely linear complex Kalman filters to handle more general models, exploiting complementary covariance for improved performance in processing improper complex signals.

Contribution

It introduces a generalized WLCKF that aligns with dual channel real Kalman filters and develops an unscented WLCKF with modified sigma points capturing full second-order statistics.

Findings

Performance improved over traditional CKF

Generalized WLCKF handles complex models better

Modified sigma points preserve complete moments

Abstract

For an improper complex signal x, its complementary covariance ExxT is not zero and thus it carries useful statistical information about x. Widely linear processing exploits Hermitian and complementary covariance to improve performance. In this paper we extend the existing theory of widely linear complex Kalman filters (WLCKF) and unscented WLCKFs [1]. We propose a WLCKF which can deal with more general dynamical models of complex-valued states and measurements than the WLCKFs in [1]. The proposed WLCKF has an equivalency with the corresponding dual channel real KF. Our analytical and numerical results show the performance improvement of a WLCKF over a complex Kalman filter (CKF) that does not exploit complementary covariance. We also develop an unscented WLCKF which uses modified complex sigma points. The modified complex sigma points preserve complete first and second moments of complex signals, while the sigma points in [1] only carry the mean and Hermitian covariance, but not complementary covariance of complex signals.